Ideals in Graph Algebras

TL;DR

This paper identifies errors in the existing graph construction methods for gauge-invariant ideals in graph C*-algebras and Leavitt path algebras, and proposes a corrected construction to accurately realize these ideals.

Contribution

The paper introduces a new graph construction that correctly represents gauge-invariant and graded ideals as graph C*-algebras and Leavitt path algebras, fixing prior inaccuracies.

Findings

Corrected graph construction for gauge-invariant ideals

Realization of ideals as graph C*-algebras and Leavitt path algebras

Improved understanding of ideal structures in graph algebras

Abstract

We show that the graph construction used to prove that a gauge-invariant ideal of a graph C*-algebra is isomorphic to a graph C*-algebra, and also used to prove that a graded ideal of a Leavitt path algebra is isomorphic to a Leavitt path algebra, is incorrect as stated in the literature. We give a new graph construction to remedy this problem, and prove that it can be used to realize a gauge-invariant ideal (respectively, a graded ideal) as a graph C*-algebra (respectively, a Leavitt path algebra).